A335942 Number of compositions of n such that the set s of parts and multiplicities satisfies s = {1..max(s)}.

1, 1, 2, 2, 3, 12, 12, 32, 51, 144, 191, 486, 679, 1487, 3149, 5909, 11637, 18630, 36928, 76431, 141009, 264784, 535057, 921105, 1774022, 3388054, 6303519, 12255373, 22527578, 43358822, 77695383, 145170435, 264722429, 527776034, 936538336, 1807344134

Offset: 0

Author

Alois P. Heinz, Jun 30 2020

Keywords

• nonn

Examples

			a(4) = 3: 211, 121, 112.
a(5) = 12: 23, 32, 113, 122, 131, 212, 221, 311, 1112, 1121, 1211, 2111.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..142 (n = 0..100 from Alois P. Heinz)

Crossrefs

Cf. A107429, A329741, A335941.

Programs

Maple

b:= proc(n, i, s, p) option remember;
     `if`(n=0, `if`(s={$0..max(s)}, p!, 0), `if`(i<1, 0, add(
      b(n-i*j, i-1, {s[], j, `if`(j=0, 0, i)}, p+j)/j!, j=0..n/i)))
    end:
a:= n-> b(n, floor((sqrt(1+8*(n+1))-1)/2), {0}, 0):
seq(a(n), n=0..35);

Mathematica

b[n_, i_, s_, p_] := b[n, i, s, p] =
     If[n == 0, If[s == Range[0, Max[s]], p!, 0], If[i < 1, 0, Sum[
     b[n - i*j, i - 1, Union@Flatten@{s, j, If[j == 0, 0, i]}, p + j]/j!,
     {j, 0, n/i}]]];
a[n_] := b[n, Floor[(Sqrt[1 + 8*(n + 1)] - 1)/2], {0}, 0];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, May 30 2022, after Alois P. Heinz *)